Firstly, due to the max flow-min cut theorem [1]}, for any multiflow (not necessarily locking) that meets the above conditions, we immediately have [2]}, [3]}
\(\int _1 {\left( {{{\vec{v}}_{12}} + {{\vec{v}}_{13}}} \right) + \int _2 {\left( {{{\vec{v}}_{12}} + \;{{\vec{v}}_{23}}} \right)} } + \int _3 {\left( {{{\vec{v}}_{13}} + {{\vec{v}}_{23}}} \right)} \le {S_1} + {S_2} + {S_3},\) 
